Use the Discriminant to Determine If the Graph of the Equation

Question

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The Answer is

3 x y = - 12

\Leftrightarrow

x y + 4 = 0

\Rightarrow

B ^ { 2 } - 4 A C = 1

, so the graph is a hyperbola. To eliminate the

x y

- term, rotate through

\phi

, such that

\cot 2 \phi = \frac { A - C } { B } = 0

\Leftrightarrow

\cos 2 \phi = 0

, so

\phi = \frac { \pi } { 4 }

.

x = X \cos \phi - Y \sin \phi = \frac { 1 } { \sqrt { 2 } } X - \frac { 1 } { \sqrt { 2 } } Y

,

y = X \sin \phi + Y \cos \phi = \frac { 1 } { \sqrt { 2 } } X + \frac { 1 } { \sqrt { 2 } } Y

. So

Y ^ { 2 } - X ^ { 2 } = 8

$3 x y = - 12 \Leftrightarrow x y + 4 = 0 \Rightarrow B ^ { 2 } - 4 A C = 1 , so the graph is a hyperbola. To eliminate the x y - term, rotate through \phi , such that \cot 2 \phi = \frac { A - C } { B } = 0 \Leftrightarrow \cos 2 \phi = 0 , so \phi = \frac { \pi } { 4 } . x = X \cos \phi - Y \sin \phi = \frac { 1 } { \sqrt { 2 } } X - \frac { 1 } { \sqrt { 2 } } Y , y = X \sin \phi + Y \cos \phi = \frac { 1 } { \sqrt { 2 } } X + \frac { 1 } { \sqrt { 2 } } Y . So Y ^ { 2 } - X ^ { 2 } = 8$

Use the Discriminant to Determine If the Graph of the Equation

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